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Let $V$ be a finite dimensional vector space over the complex field $\mathbb C$. Let $L:V\rightarrow V$ be a linear operator. Using the matrix of $L$ and the Jordan canonical form it is easy to find all the linear operators that commute with $L$. Now suppose that $H$ is a Hilbert space and let $L:H\rightarrow H$ be a continuous linear operator. There is some method to determine all the continuous linear operatores that commute with $L$? |
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Well ... yes if $L$ is normal (meaning $LL^* = L^*L$; in particular, if $L$ is self-adjoint). Assuming that $H$ is separable, we have a structure theorem which says that $H$ is isomorphic to the $L^2$ sections of a bundle over $[-\|L\|, \|L\|]$ whose fibers are Hilbert spaces, in such a way that $L$ goes to multiplication by $x$. The operators that commute with $L$ are then morally just the operators which preserve each fiber, though one has to be a little careful with measurability issues when making this precise. If $L$ is not normal then at least you can say any weak operator limit of polynomials in $L$ commutes with $L$. But I don't know if you can say much more than that in general. |
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I am far from being an expert, but there is a list of results for special cases in a book by Radjavi and Rosenthal, especially in Chapter 9. |
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